Frustum of Cone

The frustum of a cone is the part of the cone with its base that is left after the cone is cut by a plane that is parallel to its base. A frustum is defined for cones and pyramids and is also referred to as a truncated shape. There are different types of frustums depending upon the shape from which it is obtained.

• Frustum of cone
• Frustum of a triangular pyramid
• Frustum of a square pyramid, etc.

Let us learn more about the frustum of cone along with its definition, properties, volume, surface area, solved examples, and practice questions.

The frustum of a cone is the part of the cone without vertex when the cone is divided into two parts with a plane that is parallel to the base of the cone. Another name for the frustum of a cone is a truncated cone. Just like any other 3D shape, the frustum of a cone also has surface area and volume. We will see the formulas to find them in the upcoming sections. We can see how a frustum of cone can be formed in the figure below.

The net of any shape is a combination of two-dimensional shapes that are obtained by opening three-dimensional shapes. i.e., the net of a frustum when folded up gives the corresponding frustum. The net of a frustum of a cone has two circles corresponding to its two circular bases. Here is the net of the frustum of cone.

The properties of a frustum of a cone are derived by the way it is obtained from a cone. Here are the properties of a frustum of a cone.

• The frustum of a cone doesn't contain the vertex of the corresponding cone but contains the base of the cone.
• The frustum of a cone is determined by its height and two radii (corresponding to two bases).
• The height of the frustum of cone is the perpendicular distance between the centers of the two bases of the frustum.
• If the cone is a right circular cone, then the frustums formed from it also would be right-circular.

The volume of frustum of cone is the amount of space that is inside it. Just like the volume of any other shape, the volume of the frustum of cone is also measured in cubic units such as m³, cm³, in³, etc. Consider a cone of base radius R and height H + h. Assume that a frustum of a cone of height H with the large base radius 'R' and small base radius 'r' is formed from the cone. Let L and L + l be the slant heights of the frustum and the cone respectively. Then the volume of the frustum of the cone can be determined by one of the following formulas. If you are curious to know the detailed derivation of these formulas, click here.

Volume of frustum of cone = πh/3 [ (R³ - r³) / r ] (OR)

Volume of frustum of cone = πH/3 (R² + Rr + r²)

Note: Here π is a constant whose value is 22/7 (or) 3.141592653...

The surface area of frustum of cone is obtained by adding the areas of all of its faces. Since it is an area, it is measured in square units like m², cm², in², etc. The frustum of cone has two types of areas.

• Curved Surface Area (CSA) (or) Lateral Surface Area (LSA): This is the area of the curved surface of the frustum.
• Total Surface Area (TSA): This is the sum of areas of all its faces (i.e., CSA + Sum of areas of the circular bases).

Consider a cone of slant height L + l, height H + h, and base radius R. Assume that a frustum of height H, a small base radius 'r', a large base radius 'R', and slant height L is formed from the cone. Here are the formulas for curved surface area (CSA) and the total surface area (TSA) of a frustum of a cone. If you are curious to know the detailed derivation of these formulas, click here.

The curved surface area (CSA) of the frustum of the cone is,

CSA (or) LSA of frustum of cone = πl [ (R² - r²) / r ] (OR) πL (R + r)

The base areas of the frustum of the cone are obtained by using the area of a circle formula, i.e., the base areas are πR² and πr². So the sum of base areas is π (R² + r²). Thus, the total surface area (TSA) of the frustum of the cone is,

TSA of frustum of cone = πl [ (R² - r²) / r ] + π (R² + r²) (OR) πL (R + r) + π (R² + r²)

Important Notes

1. Here π is a constant whose value is 22/7 (or) 3.141592653...
2. The relation between the base radii R and r, slant height (L), and height (H) of the frustum of the cone (by the Pythagoras theorem) is, L² = H² + (R - r)². This is very helpful while solving the problems that are given with slant height.

Related Topics

Here are a few topics that are related to the frustum of cone. Check them out.

• Cone
• Frustum
• Volume of Frustum
• Surface Area of Frustum

What is Frustum of Cone?

Take a cone, cut it using a plane such that the plane is parallel to the base of the cone. Then the cone is divided into two parts. Among them, the part of the cone without vertex is called the frustum of the cone.

What are Frustum of Cone Formulas?

Here are the formulas to find the surface area and volume of a frustum. In these formulas, the frustum of radii 'R' and 'r', height 'H', slant height L is formed from a cone of base radius 'R', height 'H + h', and slant height 'L + l'.

• Volume of frustum of cone, V = πh/3 [ (R³ - r³) / r ] (OR) πH/3 (R² + Rr + r²).
• Total surface area of frustum of cone = πl [ (R² - r²) / r ] + π (R² + r²) (OR) πL (R + r) + π (R² + r²).

What are Properties of Frustum of Cone?

A frustum of cone has two circular bases and hence two base radii. The distance between the centers of the two bases is the height of the frustum. If the cone is a right circular cone, then every frustum formed from it is also a right circular frustum. These properties of the frustum of cone are derived from its definition.

What is the Surface Area of Frustum of Cone?

Just like a cone, the frustum of cone also has curved surface area (CSA) and total surface area (TSA). If 'R' and 'r' are the base radii, 'L' is the slant height, and 'H' is the height of the frustum that is formed from a cone of slant height 'L + l' , then:

• CSA of frustum of cone = πl [ (R² - r²) / r ] (OR) πL (R + r)
• TSA of frustum of cone = π (R² + r²) + πl [ (R² - r²) / r ] (OR) πL (R + r) + π (R² + r²)

What is the Volume of Frustum of Cone?

The volume of the frustum of cone is the number of unit cubes that can be fit into it. If 'R' and 'r' are the base radii, 'L' is the slant height, and 'H' is the height of the frustum that is formed from a cone of slant height 'L + l' and height 'H + h', then the volume (V) of the frustum of a cone is calculated using one of the following formulas.

• V = πh/3 [ (R³ - r³) / r ] (OR)
• V = πH/3 (R² + Rr + r²)

What is the Surface Area of Frustum of Cone Using Height?

Assume that a frustum of radii 'R' and 'r', height 'H', and slant height L is formed by a cone of base radius 'R', slant height 'L + l', and height 'H + h'. By Pythagoras theorem, the relation between the slant height (L), height (H), and base radii R and r of the frustum of the cone is, L² = H² + (R - r)². We can use this formula to calculate one of the unknown values among 'r', 'R', and 'H' and then we can find the total surface area of the frustum using the formula πL (R + r) + π (R² + r²).

What is the Volume of Frustum of Cone Using Slant Height?

Let us consider a frustum of radii 'R' and 'r', and height 'H' that is formed by a cone of base radius 'R' and height 'H + h'. By Pythagoras theorem, the relation between the height (H), slant height (L), and base radii R and r of the frustum of the cone is, L² = H² + (R - r)². This formula can be used to calculate one of the unknown values among 'r', 'R', and 'H'. We can then find the volume (V) of the frustum of cone is found using the formula: V = πH/3 (R² + Rr + r²)